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250                       ANALYTICAL MECHANICS
DISCUSSION.  The energy lost during the impact equals the kinetic energy of projection, as would be expected from the conservation of energy.
When e = 1, VQ = 0 and TI = 0. In other words when the ball is perfectly elastic it will rise to the height from which it is dropped. The entire kinetic energy is transformed, during the impact, into potential energy and back to kinetic energy without any loss.
When e = 0, v0 - co and TI = <x>, that is, if the contact is perfectly inelastic no value of the velocity of projection will enable the ball to rebound after the impact.
1.   Show that when two perfectly elastic spheres of equal mass collide centrally they exchange velocities.
2.  A ball of mass mi, impinging directly on another ball of mass m^ at rest, comes to rest.   Show that mi = e-m*.
3.  Two perfectly elastic balls collide directly with equal velocities. The relation between their masses is such that one of them is reduced to rest.   Find the relation.
4.  A ball which is dropped on a horizontal floor from a height h reaches  a height equal to I h at the second rebound.   Find the coefficient of restitution.
6. A metal patched bullet strikes a wall normally with a velocity of
1200__--   With what velocity will it rebound if e = 0.4?
6. Show that if two equal balls collide centrally with velocities ^- v
1  e
;and v, the one which has the former velocity will come to rest. *   7. A bullet strikes a vertical target normally and rebounds.   Find the relation between the distances of the foot of the target from the rifle and from the place where the bullet strikes the ground.
8.  Two perfectly elastic equal balls collide with velocities inversely as ".their masses.   Find the velocities after collision.
9.   Two billiard balls collide centrally with velocities of 8 feet per 'second and 16 feet per second.   Supposing e  0.8, find the final velocities.
10.  A ball is dropped from the top of a tower, at the same instant that another ball of equal mass is projected upward from the base of the tower, with a velocity just enough to raise it to the top of the tower.    Show that if the balls collide centrally the falling ball will rise, on the rebound, to a
height - (3 + e2) above the ground, where h is the height of the tower.